DISCLAIMERS for anyone who actually looks at this post:
this is not real lmao. i am one person who hasnât done physics by hand since college, using three equations to define interstellar travel and many assumptions which do not hold up as soon as you do anything Real - please do not take this work as hard confirmation of anything, it was merely done to satisfy my own curiosity and have fun given the limited information we have. it all falls apart under a light breeze. i know that adding a consequence for rocky giving grace fuel was done for the drama of the scene/to enhance the emotional core of the movie - but i find it scientifically interesting anyways! that said, i WOULD be curious to know if i made any critical math errors or if anyone has tried to solve this problem elsewhere (again for the sake of my own curiosity)! feel free to send me an ask or smth if thatâs the case!
when rocky says âi go home six years slowerâ in the movie, he is referring to time spent in travel per his own moving reference frame, not eridâs stationary reference frame
before meeting grace, rocky had originally expected his return trip to erid to take the same amount of time as his trip to tau ceti (so heâd travel at the same speed)
the amount of fuel rocky needs/uses for exiting/reentering eridâs orbit, entering/exiting tau cetiâs orbit, and performing station keeping around tau ceti is negligible compared to the amount of fuel needed for interstellar travel (thus ignored)
after rocky offers grace fuel, he plans on having effectively zero fuel left when he gets back to erid
dV = 2V, since this is a somewhat-accepted method to roughly calculate dV for interstellar trips in fictional situations lmao
the blip-a cannot throttle its engine(s)
SETUP: (sorry for my crappy scans throughout, i donât hate myself enough to put this all in latex lmfao)
rockyâs claim that giving up 2e6 kg of astrophage would increase the duration of his return trip to erid by 6 years, in addition to eridians not knowing about relativity (thus thinking that rocky wouldâve spent more fuel getting to tau ceti than he actually did), implies to me that the blip-a and her crew did not actually have enough fuel, from the eridiansâ perspective, to return to erid after the mission to tau ceti. i have been Extremely Curious about this since watching the movie and wanted to categorize the problem as best i could to understand rockyâs situation.
to start off with, i wanted to use as few numbers from the book/weirâs calculations as possible (while still using Enough to make sense of things ofc). this is because the fuel situation on the blip-a is obviously different between the book and movie (book rocky has a ton of extra fuel; giving some to grace has no consequence). to keep this minimal, i only used the numbers the movie provided (2e6 kg, 6 years), the distance between tau ceti and erid (which is like. an actual IRL physical distance), the dry mass of the blip-a, and the duration of rockyâs trip from erid to tau ceti from rockyâs perspective.
so, in terms of these selected values and using the drt and time dilation equations, we can find both 1) the duration of the blip-aâs actual trip from erid to tau ceti, from eridâs perspective, and 2) the duration of rockyâs planned return trip from tau ceti to erid after giving grace fuel, from eridâs perspective:
with this information, we can solve for Ve in the rocket equation. Ve (the exit velocity of fuel from an engine) is going to be characteristic of the astrophage fuel and blip-aâs engine setup so long as there is no throttling going on, so this will be consistent for all of the blip-aâs trips:
next we want to plot this. Ve values which exceed c (the speed of light) are obviously invalid, so theyâre marked as black rather than green in the plot. for a range of fuel-mass-for-return-trip-after-offering-grace-fuel values (shown as a factor of the blip-a dry mass), we can see that Ve approaches 6.7e4 at m3/m0 = 1000. iâm going to use Ve = 7e4 moving forward.
before using this value directly, weâll also want to use the rocket equation to solve for the total initial blip-a fuel mass in terms of known values/values weâre varying:
with Ve, m1, the rocket equation, and the drt equation, we can figure out how much fuel the eridians expected the blip-a to have - based on their initial incorrect estimate of how long the trip to tau ceti would take - for its return trip home:
now we plot this, varying the m3-to-m0 fuel ratio as well as Tbad. any points on this plot which have a Y value which is less than zero are marked in black, since they leave the blip-a with no fuel for the return trip (and might not even get it to tau ceti in the first place):
this is all CRAZY to me considering that the blip-a was expected to reach tau ceti in 6.6 years with 31 times its dry mass in fuel, according to weirâs calculations - that is solidly impossible for the movie!! the blip-a wouldâve needed a LOT more fuel to even make it to tau ceti in the first placeâŠ
blakekathryn
theartofmadeline
s-u-f-j-a-n-s-t-e-v-e-n-s
